New formulation for the normal gravity field

The gravity field of a level oblate spheroid is formulated in this study using various coordinate systems. From the viewpoint of physical characteristics, it is assumed that this ellipsoid of revolution encloses mass, rotates with constant angular velocity and is a level (or equipotential) surface of its own gravity field. First, a spheroidal coordinate system and spheroidal harmonics are introduced. An exterior Dirichlet boundary-value problem is solved to determine the gravitational potential. As a result, the gravity potential is calculated completely and uniquely outside of the ellipsoid. Its closed form is then given in Cartesian, spherical, and geodetic coordinates. The gravity vector is calculated from the gravity potential in the exterior space and on the surface of the level ellipsoid. Additionally, the classical theorems of Clairaut, Pizzetti, and Somigliana are presented. Second, because the Earth's ellipsoid deviates slightly from a sphere, series expansions in terms of eccentricities for the normal gravity field are provided. Finally, the field is expanded in terms of spherical harmonics, which are useful for interpretations and calculations.